Strong first order topological phases are classified in the tenfold way [1]}, [2]} and do not require any crystalline symmetries for their protection.
They may be realized in systems without a lattice structure where the dimensionality is only enforced by the locality of the Hamiltonian.
Crystalline symmetries however may prohibit the existence of first order topological phases. A typical example is that the presence of a mirror symmetry requires the Chern number in a plane perpendicular to the mirror plane to vanish.
